Twisted bilayer zigzag-graphene nanoribbon junctions with tunable edge states

Stacking two-dimensional layered materials such as graphene and transitional metal dichalcogenides with nonzero interlayer twist angles has recently become attractive because of the emergence of novel physical properties. Stacking of one-dimensional nanomaterials offers the lateral stacking offset as an additional parameter for modulating the resulting material properties. Here, we report that the edge states of twisted bilayer zigzag graphene nanoribbons (TBZGNRs) can be tuned with both the twist angle and the stacking offset. Strong edge state variations in the stacking region are first revealed by density functional theory (DFT) calculations. We construct and characterize twisted bilayer zigzag graphene nanoribbon (TBZGNR) systems on a Au(111) surface using scanning tunneling microscopy. A detailed analysis of three prototypical orthogonal TBZGNR junctions exhibiting different stacking offsets by means of scanning tunneling spectroscopy reveals emergent near-zero-energy states. From a comparison with DFT calculations, we conclude that the emergent edge states originate from the formation of flat bands whose energy and spin degeneracy are highly tunable with the stacking offset. Our work highlights fundamental differences between 2D and 1D twistronics and spurs further investigation of twisted one-dimensional systems.


Reversible STM Tip Manipulation of ZGNRs
It is reported that the graphene nanoribbon with armchair edge has superlubricity on gold surface 1 . Here, by STM tip manipulation, we proved that the graphene nanoribbon with zigzag edge can also be easily lateral moved and manipulated on gold surface like the arm-chair case 2 . As shown in Supplementary Figure 2(b), the ribbon was firstly bent by the STM lateral manipulation in a direction indicated by the white arrow. Then, the ribbon was bent back to its original position with tip manipulation in a reversal direction as shown in 2(c) and 2(a). Furthermore, we bent the ribbon again in the direction same with 2(b) but with a smaller bending angle as shown in 2(d). From 2(a) and 2(c) we can see the ribbon has similar quality after the bending.

STM images of more TBZGNR Junctions
With the STM lateral tip manipulation technique mentioned above, we can achieve TBZGNR junctions with different twist angles. 16

Charge redistribution within the TBZGNR junction
Previous study on bilayer graphene 3 shows the finite interlayer hopping could influence the low-energy band structure of graphene. This is also true for the case of bilayer graphene nanoribbon. The interlayer hopping manifests itself as a finite bonding between the electrons of the top and bottom ribbon. Thus, the effective electron charge redistributes in the overlapping region. By DFT calculation, this effect is clearly shown in Supplementary Figure 4 where a net electron charge accumulation happened in the space between the top and bottom ribbon.

DFT results for another TBZGNR structural model, Model D
In order to further verify the theory that a TBZGNR without symmetry can support spin polarized flat band at the edge as we argued in the main text and Figure 4 we can see again a pronounced peak near zero energy is developed only near the corner of the overlapped region. This peak also originates from a new flat band near zero shown in 5(c). We can further identify the bands are spin non-degenerate with a relatively small splitting energy. All these features are similar to what we obtained for the model C in the main text. Thus, our argument that the asymmetrical van de Waals potential produces spin polarized flat bands in TBZGNR is repeated in another model structure.

Influence of the Au (111) surface state and step edge on the spectra of monolayer ZGNR
As the ZGNR used in this work is the same as that used in the previous reference 4 , the edge only terminated with C-H bond. Because we do not intentionally make a bias pulse, the C-Au bonding reported in the other reference 5 is also not the case. From the previous theory prediction 6 it is shown that the ZGNR on Au (111) still displays a magnetic edge state with antiferromagnetic coupling between the edges. The magnetization per edge C atom is about 0.22 μB which is comparable to the freestanding ZGNRs. These edge states are not observed mainly due to the strong extension of the surface state of Au (111) in the out of plane direction. From the Figure   2 in previous report 7 we can clearly see the Au (111) surface state survive even 2 Å away from the surface. The apparent height of our monolayer ZGNR is just 1.85 Å.
Thus, most of the ZGNR edge states are in the shadow of the Au (111) surface state, this is also the reason why the DOS at the monolayer edge mimics the surface state of Au (111), as already seen in the Figure 2h.
Recently, an interesting work 8 reported effective pi bonding between C and Au atoms when the ribbon edge is doped by nitrogen. However, this is not the case here as we never see an effective decoupling of the monolayer ribbon by ramping a bias sweep, also the ribbon used is not nitrogen doped.
As the bottom ribbon of the junction always lies in the vicinity of the Au (111) step edges, it is a necessary to discuss the influence of the step edges on the density of state we obtained on the TBZGNR junction. Taking one junction with a twist angle near topography image shown in Supplementary Figure 7(a). By comparing with corresponding dI/dV mapping images at energies we interested in this study, we did not find any evident density of state distribution on the Au step edges, as shown in 7(b) and 7(c). In contrast, the edge state of the top zigzag graphene nanoribbon at the junction is clearly demonstrated, as highlighted by the yellow arrows. Thus, it concludes that the step edge of the Au (111) will not give additional influence on our data analysis regarding the energies we are interested.

Exclusion of out-of-plane bending and lattice distortion effects
We estimate the out-of-plane bending effect of the top ZGNR by taking the line profile across the TBZGNR junction, as demonstrated in Supplementary Figure 10 On the other hand, the lattice distortion in graphene system can also introduce some important physical effect such as pseudo-magnetic field. Compared to the previous reports where the strain mainly due to a designed structure confinement by the substrate 10,11 , the fabrication process of TBZGNR does not introduce any additional strain, as: 1) the Au surface is flat and no substrate template effect; 2) the end of the top ribbon is free, strain due to confinement can be ignored. There are some possibilities that the Van der Waals and gravity can introduce some lattice distortion at the TBZGNR junction edge, However, from some simple calculations we can show that these two effects can also be excluded, as shown below: We consider two extreme cases: It is demonstrated experimentally that 1%-2% strain in graphene can only introduce 0.7 T pseudo-magnetic field 13 . So, in our case there is no additional strain and related pseudo-magnetic field induced by both gravity and van der Waals force. This is also demonstrated in Figure 3 that no corresponding symmetric Landau Levels found in experiment and DFT calculation.
There is another evidence that the near zero energy bound state does not originate from strain effect. The dI/dV mapping image at -40 mV in main text Figure 2g shows the bound state localized at both the edges and the corners of the junction. The periodicity is the same as that of the zigzag carbon atoms, which demonstrate the near zero energy bound state is junction structure related other than strain related.

Exclusion of supercell size and a possible intersupercell interaction effect
The total energy per atom as a function of the lattice constant of the supercell were carefully tested in the calculation. As shown in the Supplementary Figure 8 We further did a calculation on a non-periodic structure for comparison as shown in Supplementary Figure 8 We also checked the PDOS at the edge of monolayer ZGNR in the vicinity of the TBZGNR junction with DFT calculation, as shown in Supplementary Figure 9. The PDOS at the monolayer ZGNR edge already demonstrating a gap like line shape, mimic that of the pristine GNR, although the corresponding carbon atom just shift 2 lattice constants from the junction corner. The featured near-zero-energy peaks observed on the edge atoms in the crossing region are obviously absent.
Thus, the possibility that the emergence of the near-zero-peak owing to the calculating size of the supercell and possible interactions between periodic supercells are excluded.

Exclusion of the influence of bright protrusions on the edge state
As there are some bright protrusions on Figure 2f and 2j, it is a necessary to check they has no relation with the edge states we discuss in the paper. By checking the dI/dV mapping in Fig. 2g we found no additional signal belongs to the protrusion shown in Fig. 2f at the edge state energy -40 mV. From the tip manipulation of the top ribbon shown in Fig. 2i-2k, we learned the bright feature on the right edge of top ribbon appears for the first manipulation (Fig. 2j) and disappears again (Fig. 2k) after manipulating back. As our tip is far from the junction during the manipulation, it is very unlikely to be adatom. These bright protrusions are also excluded to be "mousebite" type defect on the edge of the single layer ZGNR because of two facts. Firstly, the bright protrusions shown in Figs. 2f,j and Fig. 3b appear in the middle of the top ribbon edge within the junction, which means they sit on the middle of the bottom ribbon. However, the "mouse-bite" type defect appears mostly on the edge of the bottom ribbon. Secondly, as seen in Figure S4 in the supplementary information of reference 4, the STS on the "mouse-bite" type defect (indicated by red triangle) does not resemble what we have shown in Figure 3(a-c), where either clear gap feature or in-gap states were observed. Thus, we can safely exclude that the bright features in Figs. 2f,j and the insets for Fig. 3a,b are this type of defect. We propose the protrusions we observed to be a tiny stress states at the edges which, as we demonstrated already in previous section, will not lead to great impact (the top ribbon edge at the junction is still very straight and has a strain less than 1%).

Nc-AFM measurements on a 76 o TBZGNR junction together with DFT calculation
To unambiguously make a link between the structure of the junction and the It is clear that in the 4-ZGNR there are some changes on edge states but not very pronounced, while in the 8-ZGNR the edge-states changes are much more abundant than those in 4-ZGNR and in 6-ZGNR, suggesting that the wider ZGNRs will produce more complicated overlapped configurations and more abundant edge states.